Equations — Hard Practice Quiz

A Algebra cheat sheet for Equations — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. Consider a quadratic equation $kx^2 - (k+1)x + 1 = 0$. If one root is twice the other, what is the sum of the roots?

   • $\frac{3}{2}$
   • $\frac{9}{2}$
   • $\frac{1}{2}$
   • $\frac{5}{2}$

   Answer: $\frac{9}{2}$

2. A quadratic equation $x^2 + bx + c = 0$ initially has two distinct real roots. If the constant term $c$ is increased by a value $\Delta c > 0$ such that the new equation has exactly one real root, what must be true about the discriminant of the original equation?

   • The original discriminant was equal to $4\Delta c$.
   • The original discriminant was less than $4\Delta c$.
   • The original discriminant was greater than $4\Delta c$.
   • The original discriminant was equal to $2\Delta c$.

   Answer: The original discriminant was equal to $4\Delta c$.

3. Given the quadratic equation $2x^2 - 5x + 1 = 0$ with roots $x_1$ and $x_2$, what is the value of $\frac{1}{x_1} + \frac{1}{x_2}$?

   • $5$
   • $\frac{5}{2}$
   • $-5$
   • $\frac{1}{5}$

   Answer: $5$

4. The coefficients $a, b, c$ of a quadratic equation $ax^2 + bx + c = 0$ are determined by the linear equations: $2a - 4 = 0$, $3b + 9 = 0$, and $c - 1 = 0$. What is the nature of the roots of this quadratic equation?

   • Two distinct real roots
   • One real root (repeated)
   • Two complex conjugate roots
   • No roots

   Answer: Two distinct real roots

5. For a quadratic equation $ax^2 + bx + c = 0$ with roots $x_1$ and $x_2$, express $(x_1 - x_2)^2$ in terms of $a, b, c$.

   • $\frac{b^2 - 4ac}{a^2}$
   • $\frac{b^2 - 2ac}{a^2}$
   • $\frac{b^2 + 4ac}{a^2}$
   • $\frac{b^2 - ac}{a^2}$

   Answer: $\frac{b^2 - 4ac}{a^2}$

6. For what range of values of $k$ does the quadratic equation $x^2 - (k+2)x + (k+5) = 0$ have no real roots?

   • $-4 < k < 4$
   • $k < -4$ or $k > 4$
   • $k = 4$
   • $-2 < k < 2$

   Answer: $-4 < k < 4$

7. If the sum of the roots of a quadratic equation is $5$ and the product of the roots is $6$, what are the roots of the equation?

   • $2, 3$
   • $-2, -3$
   • $1, 6$
   • $-1, -6$

   Answer: $2, 3$

8. Consider the equation $mx^2 + (2m-1)x + m-2 = 0$. If $m$ is the solution to the linear equation $3m - 6 = 0$, what is the nature of the roots of the quadratic equation?

   • Two distinct real roots
   • One real root (repeated)
   • Two complex conjugate roots
   • Cannot be determined

   Answer: Two distinct real roots

9. If $x_1$ and $x_2$ are the roots of the equation $x^2 - px + q = 0$, which of the following is a quadratic equation whose roots are $\frac{1}{x_1}$ and $\frac{1}{x_2}$?

   • $qx^2 - px + 1 = 0$
   • $x^2 - qx + p = 0$
   • $px^2 - qx + 1 = 0$
   • $x^2 - px + q = 0$

   Answer: $qx^2 - px + 1 = 0$

10. A quadratic equation $x^2 + bx + c = 0$ has complex roots $2+3i$ and $2-3i$. What are the values of $b$ and $c$?

    • $b = -4, c = 13$
    • $b = 4, c = 13$
    • $b = -4, c = -5$
    • $b = 4, c = -5$

    Answer: $b = -4, c = 13$